AH!
the input to game needs to include the path to the class, and the . with repeat and _C there. That info was buried pages back in my search.
/Game/BlueprintsLogic/GameModes/L_Deathmatch.L_Deathmatch_C
But to make it easier, you can go to your DefaltGame.ini and add something like this:
[/Script/Engine.GameMode]
+GameModeClassAliases=(ShortName=“DM”,GameClassName="/Game/BlueprintsLogic/GameModes/L_Deathmatch.L_Deathmatch_C")
that creates an alias to the mode so you can just use game=DM in that instance and it loads the proper mode when loading the level.